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Diffusion of the random Lorentz process in a magnetic field

Christopher Lutsko, Balint Toth

Abstract

Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method established by the authors, we show that this 'magnetic Lorentz gas' satisfies an invariance principle in an intermediate scaling limit. That is, we apply the low-density (Boltzmann-Grad) limit and simultaneously take the limit as time goes to infinity, then prove convergence of the rescaled trajectory to a Brownian motion in this limit.

Diffusion of the random Lorentz process in a magnetic field

Abstract

Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method established by the authors, we show that this 'magnetic Lorentz gas' satisfies an invariance principle in an intermediate scaling limit. That is, we apply the low-density (Boltzmann-Grad) limit and simultaneously take the limit as time goes to infinity, then prove convergence of the rescaled trajectory to a Brownian motion in this limit.

Paper Structure

This paper contains 17 sections, 8 theorems, 76 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The processes
  3. The magnetic Lorentz process
  4. The limit process
  5. The Markovized version of the MLP
  6. Magnetic Lorentz process
  7. Main technical result
  8. Proof
  9. Decomposition into legs
  10. Döblin condition
  11. Breaking $(\psi_n)_{n\ge 1}$ into packs
  12. Green's function estimates
  13. Inter-leg mismatches
  14. Intra-leg mismatches
  15. A geometric lemma
  16. ...and 2 more sections

Key Result

Theorem 1

Let $T(\varepsilon)$ be such that $\lim_{\varepsilon\to 0} T(\varepsilon) =\infty$, $\lim_{\varepsilon\to 0} \varepsilon\left|{\log{\varepsilon}}\right| T(\varepsilon)=0$. Then, under the Boltzmann-Grad limit BG-Limit, where $\Rightarrow$ denotes weak convergence of trajectories in $C([0,1]\to\mathbb R^2)$, and, on the right hand side, $\alpha$ is a Bernoulli random variable with distribution $\m

Figures (4)

  • Figure 1: Here we give two examples of the magnetic Lorentz gas travelling through the same array of scatterers. Collision (a) is an example of a recollision with the 'current scatterer', while collision (b) is an example of a recollision with a 'past scatterer'.
  • Figure 2: The four possible scenarios for the magnetic Lorentz gas are shown here as described above.
  • Figure 3: We illustrate a sample trajectory of the Markovized version of the MLP at three successive times. Note that the third panel would not be possible for the physical MLP, since the scatterer is placed in a region of space which was already explored (a shadowed collision). However, the Markovized version ignores this physical complication and collides anyways.
  • Figure 4: On the left we give an example of a recollision with a past scatterer. The physical MLP (black) must respect this collision. Whereas the Markovized MLP (red) has forgotten about the past scatterer and thus ignores the recollision. On the right we give an example of a shadowed collision. Here the Markovized MLP (red) changes direction in the third collision. However this scatterer is in previously explored space, thus the physical MLP must ignore the collision and continues in its trajectory. We call this a shadowed collision.

Theorems & Definitions (12)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • Theorem 2
  • Lemma 1
  • proof
  • Proposition 3
  • Lemma 2
  • proof
  • ...and 2 more